Energetics of normal earthquakes on dip-slip faults

The problem

Earthquakes release large amounts of energy, but often only a small fraction of it (the seismic efficiency) is radiated as the seismic waves felt as shaking (McGarr, 1994; Kanamori, 2001). The rest moves, deforms and heats the crust.

The standard description of an earthquake is elastic rebound (Reid, 1911). Tectonic forces slowly strain the rock either side of a locked fault. The strained rock stores elastic strain energy. When the fault slips, that energy is released as the rock returns towards its unstrained shape. Reid developed this model for strike-slip faults, where the two sides move horizontally and gravity does no work.

Normal faults form where the crust is being stretched. In a normal earthquake the block above the fault, the hangingwall, drops. Geodetic surveys, which repeatedly measure ground elevation, record this fall (Barrientos et al., 1987). Gravitational potential energy released by the fall contributes to the energy budget of these earthquakes (McKenzie, 1978; Stewart, 1980). Our goal was to quantify that budget for a normal fault, and to find where in the crust energy is given up and taken up during the earthquake cycle.

What we did

Following Ellis et al. (2006), we built a two-dimensional finite element model of a crust 60 km wide and 25 km deep. Its sides were pulled apart at a combined 12 mm per year, or about 1.2 m per century. The model rock deforms elastically, fails by frictional sliding on small fractures near the surface, and creeps slowly at depth. The transition from brittle to ductile behaviour occurs where temperatures exceed about 350 °C.

The crust contains one fault, which dips at 50° from horizontal to a depth of 15 km. We locked the fault for 250 years with infinite friction. We then lowered its friction coefficient (the ratio of the force needed to slide the fault to the force pressing it shut) to 0.2 for 60 seconds to represent an earthquake. The model is static and does not simulate seismic waves. We report energies from the 12th cycle, once the stress pattern repeated from one cycle to the next.

Each model earthquake produced 1.2 m of vertical offset at the surface, as the footwall rose and the hangingwall dropped. This pattern is similar in shape to measurements after the 1983 Borah Peak (Idaho, magnitude 6.9) and 1959 Hebgen Lake (Montana, magnitude 7.3) earthquakes (Barrientos et al., 1987). Our offset corresponds to a smaller earthquake, about magnitude 5.8.

We balanced the work done by gravity and by the model boundaries against the change in elastic strain energy and the energy lost to creep, to sliding on small fractures, and to friction on the fault (Cooke and Murphy, 2004). We split the elastic strain energy into a volumetric part, from changes in rock volume, and a distortional part, from changes in rock shape.

What we found

In our model, the total elastic strain energy of the crust decreases between earthquakes, although it increases locally in the footwall at 8 to 10 km depth. This is because the weight of overlying rock keeps the crust in compression almost everywhere. Stretching reduces that compression and lowers the stored energy. The crust therefore does work, expending stored energy, as it extends.

During an earthquake, the footwall rebounds elastically. A drop in stress and an expansion of rock at 8 to 10 km depth release about 0.85 TJ of elastic strain energy. Of this, 0.43 TJ lifts the footwall, 0.14 TJ heats the fault by friction, and 0.28 TJ pushes on the hangingwall.

The hangingwall is driven by gravity. Next to the fault it subsides by up to 1 m at the surface. This fall releases about 1.0 TJ of gravitational potential energy. Together with 0.1 TJ of distortional strain energy and the 0.28 TJ from the footwall, this energy compresses the crust. It creates about 1.3 TJ of volumetric strain energy, particularly at 6 to 10 km depth where slip on the fault stops, and heats the fault by another 0.14 TJ.

As the two fault blocks move apart, each pushes horizontally against the crust beyond it. This compresses most of the upper crust. The increase in volumetric strain energy there is about 50% larger than the simultaneous release of distortional strain energy. The net elastic strain energy of the crust therefore rises during an earthquake.

We use a spring-and-block model to explain why the hangingwall slips. Gravity pulls the block down the fault, and horizontal compression pushes it up the fault. Stretching reduces the horizontal push and leaves the gravitational pull unchanged. When the net down-dip force, which pulls the block down the fault, exceeds friction, the block slides down, recompresses the crust and restores the balance. Gravitational potential energy is present throughout the earthquake cycle but does work only when the fault slips. We suggest tectonic stretching shifts the balance of forces on a crust that is already loaded, and the energy of normal faulting comes from gravity and stored elastic strain.

Limitations

The model is a single two-dimensional cross-section through one fault. The model earthquake is smaller than the Borah Peak and Hebgen Lake events. The difference may reflect greater stress build-up on those faults or different rock stiffness (Gudmundsson, 2004). The model also neglects seismic waves and the way fault friction changes with slip speed (rate-dependent friction). Estimating radiated energy and slip after an earthquake would require a model that includes them.

Glossary

A normal fault is a fault where the crust is being stretched and the upper block slides down the fault plane.

The hangingwall is the block of crust above a dipping fault, and the footwall is the block below it.

Elastic strain energy is energy stored in rock that is compressed or distorted and can be recovered when the rock returns to its original shape.

Volumetric strain energy comes from changes in rock volume, and distortional strain energy comes from changes in rock shape.

Gravitational potential energy is energy a mass has because of its height. It is released when the mass moves down.

Elastic rebound is the release of stored strain energy when a fault slips and the surrounding rock relaxes.

The brittle–ductile transition is the depth below which rock deforms mainly by slow creep.

Seismic efficiency is the fraction of an earthquake's energy that is radiated as seismic waves.

A geodetic survey repeatedly measures the position and elevation of points on the ground.

A friction coefficient is the ratio of the force needed to slide a fault to the force pressing it shut.

A fault dips at the angle it is inclined below horizontal. Down-dip means down the slope of the fault.

Rate-dependent friction is fault friction that changes with the speed of slip.

A finite element model solves the equations of deformation by dividing the crust into many small elements.

A terajoule (TJ) is one million million joules.